Luschgy, H., Mussmann, D. and Yamada, S.Osaka J. Math.25 (1988), 795-803

MINIMAL L-SPACE AND HALMOS-SAVAGE CRITERIONFOR MAJORIZED EXPERIMENTS

H. LUSCHGY*, D. MUSSMANN, AND S. YAMADA

(Received July 29, 1987)

1. Introduction

An experiment 8 is a triplet (X, Jl, ,ίP), where S is a non-empty set ofprobability measures on a σ -field Jl of subsets of a set X. ca(JL) denotes thespace of all bounded signed measurs on JL. The closed vector sublatticeLm(6) of ca(JL) generated by S is called the minimal L-space of the experi-ment 8 (Le Cam [8], p. 41). 8 is said to be majorized if there exists a measureμ on JL such that each P^3? has a density with respect to μ. In this case,μ is called a majorizing measure for Q. The class of majorized experimentsincludes the weakly dominated experiments, where μ is localizable (see Mussmann[12]), the Σ-finite experiments (see Le Cam [8], p. 13 and p. 667), where μ isdecomposable, the semi-decomposable experiments (see Luschgy and Mussmann[10]), and the discrete experiments, where μ is the counting measure on 2X (seeBasu and Ghosh [1]).

For v*Ξca(JL)+, a set S in Jhvhich satisfies v(Sc)=0 and P( Π S)<z> for allPξ=3? is called an <?-support of v. 8 is majorized if and only if each PGΞί? hasan (^-support (cf. Diepenbrock [2], Lemma 9.3, Ramamoorthi and Yamada [15],Proposition 1, or Luschgy and Mussmann [9], Theorem 1). Throughout thepresent paper we assume that 8 is majorized. For a set H of measures onJL, put N(H}= {A^JL: v(A)=Q for all v^H}. If {/*,-, i<=I} is a family of JL-measurable functions, then cr(hiyi^I)\/N(H) denotes the smallest sub-cr-field(subfield, for short) of Jl which contains N(H) and for which each λt , ie/, ismeasurable. A subfield IB of JL is said to be PSS (pairwise sufficient containingsupports) for 6 if IB is pairwise sufficient for 8 and each Peί? has an <?-supportbelonging to S. An equivalent majorizing measure μ is called pivotal measurefor 6 if the following condition is satisfied: a subfield £B of JL is PSS for 8 ifand only if each P^S has a ^-measurable /^-density (cf. Ramamoorthi andYamada [15]). Obviously, μ is pivotal if and only if

dμ

*) Supported by a Heisenberg grant of the Deutsche Forschungsgemeinschaft

796 H. LUSCHGY, D. MUSSMANN AND S. YAMADA

is smallest PSS. If S is majorized by a σ-finite measure then the sufficiencycriterion of Halmos and Savage [6], Theorem 1, implies that each equivalentfinite majorizing measure of the form Σ cnPn with cn^0 and Pn^3? is pivotal.

«From this theorem the Neyman factorization theorem easily follows. In orderto prove extensions of these results for arbitrary majorized experiments, pivotalmeasures have been used by Ghosh et al. [5] and by Ramamoorthi and Yamada[15]. In [20] pivotal measures have been applied to construct common condi-

tional probabilities in an extended form. We shall show that pivotal measuresare closely related to maximal orthogonal systems in Lm(β).

An orthogonal system IF in a vector lattice V is a subset of F+\{0} suchthat u/\v— o for all distinct members u and v of W. If DdLm(β) is a maximalorthogonal system, we define a measure VD on Jl by vD(A)— sup {Σ w(A): FdD,

we-F

F finite}. Notice that each maximal orthogonal system of Lm(S) is also amaximal orthogonal system of the L-space of 6 and therefore VΌ is an equivalentmajorizing measure for S (Luschgy and Mussmann [9], Theorem 1, see alsoTorgersen [19], p. 10). We shall prove the following results: σ(dP\dvD,P^&}V N(S>) is a smallest PSS subfield and a pairwise smallest sufficient subfield inthe sense of [5]. This implies that VD is pivotal. Conversely, each pivotalmeasure is of the type VD. Lm(β) can be characterized as the set of all measures

on <JL having σ(dP\dvΌ, P€Ξ£P)V N(3?) -measurable densities with respect to VD.This generalizes a result by Torgersen [18], p. 47.

Furthermore, we discuss the relation between maximal orthogonal systemsin Lm(6) and maximal decompositions of X which have bεen used in the liter-ature to prove the existence of pivotal measures (cf. Ramamoorthi and Yamada[15]). We need some more notations. Let μ be a measure on Jl. L\μ) denotes the space of all μ-integrable functions. If /EΞL^μ), then/ μ, is the bound-ed signed measure on Jl with μ-density/. Set L(μ)—{f μ:f^L\μ)}. Themap from L\μ) onto L(μ) which carries f^L\μ) into f μ is an isometricvector lattice isomorphism. This is easily seen by means of the Radon-Nikodymtheorem since {/>0} has σ-finite μ-measure for each/GΞL^μ).

2. Auxiliary Results

Put β*=(X, Jl, 5>*) where 3?* is the set of all probability measures inLm(β). In the following we shall see that in some situations β can be replacedwithout loss of generality by <?*.

Proposition 2.1. Suppose ^dJί is a subfield. Let W denote the set of allw(Ξca(Jί) of the form w=f»(Σ 2~nPn) where f is ^-measurable and Pne£P. Thefollowing assertions hold :

a) If each P^S* has an 6 -support belonging to IB, then each w^W has an β-

MINIMAL L- SPACE AND HALMOS- SAVAGE CRITERION FOR MAJORIZED EXPERIMENTS 797

support which belongs to IB.b) // 3$ is pairwise sufficient for 5?, then W is a closed vector sublattίce of ca(Jΐ)

and <B is pairwise sufficient for the set of all probability measures from W.

Proof.

a) If / is ^-measurable, ^=/ (Σ2"nPΛ), w^ca(Jΐ), and Tn<=:& is an €-

support for Pn for each positive integer n, then

{/>0}n(U T.)e£is an (S-support for w.

b) If 3ί is pairwise sufficient for 5>, it is also sufficient for each dominatedsubset of 5*. Therefore, by a theorem of Halmos and Savage [6], Theorem1, we can assume that dP/d(Σ2~nPn) is ^-measurable whenever P^3> is

absolutely continuous with respect to Σ 2~nPn. From this we obtain that»

for each sequence (wm) in W there is a sequence (fm) of ^-measurablefunctions and a sequence (Pn) in & such that Wm= /m (Σj2~*Pw) for all m.

By means of these representations it is easily shown that W7 is a closed vectorsublattice. Furthermore, we see that it only remains to prove that £B is suff-cient for subsets of probability measures w^ W of the form W= f (Σ 2"nPn)

where/ is ^-measurable and the sequence (Pn) is fixed. For such a w^. Wwe get

\Λda=\B JB

= E.(lA\$)dw for all AtΞjl andJB

where EΦ(1A\1B) is a common conditional expectation for the sequence (Pn)

E.(lA\&)=EPn(lA\3)Pn-a.e. for all n. Π

Corollary 2.2. Suppose 3JC.JI is a sub field. Then the following assertions

hold:a) If each Pe.ίP has an 6 -support belonging to J3, then each Q^S?* has an

6-support belonging to J$.b) If IB is pairwise sufficient for Qy then £B is pairwise sufficient for G*.

An inspection of the proof of Proposition 2.1 shows that Corollary 2.2.b holds if"pairwise sufficient" is replaced by "sufficient". Notice that here and in Prop-osition 2.1.b and Corollary 2.2.b we do not use the assumption that 6 is ma-jorized, From [9], Lemma 1, we see that £* is majorized by a measure μwhenever 8 is majorized by μ.

Lemma 2.3. Suppose Q is majorized by μ. Then we have

798 H. LUSCHGY, D. MUSSMANN AND S. YAMADA

Proof. The set of all f&L\μ) such that / is <r(dP/dμ,

measurable is a closed vector sublattice of L\μ). Because of the vector lat-tice isomorphism between L\μ) and L(μ) (see Section 1), the proposition easilyfollows. Π

Now we need a lemma which we shall use for the calculation of the den-sities if the majorizing measure is of the form VD (see Section 1). If V is an

L-space, we define πx(y)= sup(y/\nx) for all x, yG V*\ πx(y) is the projection

of y onto the band generated by x ([16], Proposition II.2.11 and Corollary 2).

Note that for every L-space there exist maximal orthogonal systems by Zorn'slemma.

Lemma 2.4. Suppose V is an L-space. Then the following assertions hold:a) If (xiy ι € Ξ l ) is an increasing net in V+ with supH^H^oo, then lim#, exists

and lim #f = sup xf.i i

b) If D is a maximal orthogonal system in V andy^ V+, then

y = sup Σ τcu(y) = lim Σ ιt*(y)A «eA Λ «eΛ

where Λ ranges through finite nonempty subsets of D. The set {u^D: πu(y) =t= 0}is countable.

Proof.a) See [3], proof of Theorem 26 B.b) By [7], Lemma 3.5, we get

Then a) implies

sup Σ πu(y) = lim Σ πu(y) .Λ weΛ A «eA

By the Riesz decomposition theorem ([16], Theorem II.2.10), the band in Vgenerated by D is equal to V. From [16], Proposition II.2.11, we get

&wpππu(y)=y. If τru(y)ΦQ for all u from an uncountable subset of Z>,Δ «eA

then there is an £>o such that {u^D: \\τtu(y)\\*£6} is infinite. Because

l^ Σ llTΓuί^)!! f°r all Λ) we get a contradiction. Π

If u,v^Lm(£)+, then πυ(u)^Lm(8) by Lemma 2.4.a. Using the vectorlattice isomorphism between L\μ) and L(μ) with ^t=«+£;, we get

MINIMAL L- SPACE AND HALMOS- SAVAGE CRITERION FOR MAJORIZED EXPERIMENTS 799

/ \ (dU Λ \

μv(u) = \—- itiv/dμxύrμ -

Proposition 2.5. Suppose DdLm(6] is a maximal orthogonal system. Then

for each v^LJG}* there is a countable subset D'dD such that Σ ls " - M «7 /- . , weU' W dW

density of v with respect to VΌ.

Proof. By Lemma 2.4.b, there is a countable subset D'cD such that v=

Σ κw(v). Using properties of the (^-supports Sw of «?, we see thatwe/?7

,__ v Λ dπw(v)J ^-*f Sw jWGD' (fan

is a density of v with respect to VD. Π

Proposition 2.5 is essentially known. The above form of the density is

given by Torgersen [19], p. 10, for v

EXAMPLE 2.6.

a) Suppose 8 is majorized by a σ-finite measure. Then there is a majorizing

measure of the form v= Σ2~MPM, Pn^3?. The set D={v} is a maximal

orthogonal system in Lm(6) and V=VD.

b) If <_Λ! is the power set of X and if 3> contains all Dirac measures, then the

subset D of all Dirac measures is a maximal orthogonal system in Lm(8)

and VD is the counting measure.

3. Main Results

In the situation of Example 2.6.a it is known that σ(dP\dvD, P^^VN^)is a smallest sufficient subfield. For an arbitrary majorized experiment we shallshow in Theorem 3.1 that a subfield of this form is not dependent on the special

maximal orthogonal system D and that it is smallest PSS.

We define a subfield JHm^.J[ by

where DaLm(6} is a maximal orthogonal system. We use the terms "pairwise

smallest sufficient" and "smallest pairwise sufficient containing supports'' smal-lest PSS, for short) in the sense of [5].

Theorem 3.1. The subfield <Λm is pairwise smallest sufficient and smallest

PSS for 6. Especially, VD is a pivotal measure for each maximal orthogonal

system DdLm(β).

800 H. LUSCHGY, D. MUSSMANN AND S. YAMADA

Proof. Without loss of generality we assume 6=6* (see Proposition 2.1).Obviously, <JLm contains an ^-support for each PeίP. Next we show that <Am

is pairwise sufficient. Suppose P1? P2e5>. Put μ=P1+P2^Lm(6). There arecJ[w-measurable versions of dP^dμ since

dPt dPn (dμ\l

-j-1 = -j- 1 {dHd-»D>*}\ -f~ ) μ-a e .aμ avD ^άv^

Thus JLm is sufficient for {P1? P2} because of [6], Theorem 1. It remains toinvestigate the minimality of Jlm. Let SdJl be a pairwise sufficient subfield.For Pe<? there is a countable subset D'cD such that P= Σ ar«(P) (see

«;eι>'

Lemma 2.4.b). Let κ^ca(JL) be of the form /e= Σ V<>> cw^ΰ. Since 6=6*toez/

and since in the dominated case pairwise sufficiency implies sufficiency, S issufficient for

iMl^w: w^D'} U flk^ll-W: ^r^ΦO, eϋ^Z)'} .

By [6], Theorem 1, we may assume that dπw(P)ldκ and dw/d/c are <5-measurable

for all we D'. Furthermore,

dP _ _

Hence dP\dvD is <5 V AΓ(/c)-measurable. For fixed Ply P2^S we may suppose thatP1? P2</c holds in the above calculation. Therefore dP\dvD is <SV N(Pλ+P2)-measurable for all P:, P2eP, and <Jlm is pairwise smallest sufficient. If S con-tains an fi-support for each PeίP, then {dv)ldvD>Q}^ScvN(S) for alland

Hence dP\dvD is cSVΛ^(5>)-measurable. We conclude that cJJw is smallest PSSand VD is a pivotal measure. Π

The existence of a smallest PSS subfield has been proved by Ghosh etat. [5], Theorem 5. A detailed discussion of the smallest PSS subfield can befound in Fujii and Morimoto [4], Theorem 5. Pairwise smallest sufficiency istreated in Siebert [17] (see also [5], Theorem 5) and using invariance considera-tions in [11]. In Theorem 3.4 we shall see that any piovtal measure can berepresented by means of a suitable maximal orthogonal system in Lm(6). Firstwe give a more concrete representation of Lm(6).

Theorem 3.2. Suppose DdLm(6) is a maximal orthogonal system. Then

MINIMAL L-SPACE AND HALMOS- SAVAGE CRITERION FOR MAJORIZED EXPERIMENTS 801

Proof. It suffices to prove that the inclusion ID holds. Because of the formof the densities given in Proposition 2.5, it is enough to show for each fixed

that for each/eL^Mw) there is a u^Lm(6) with

s1B - y - s w . .avD

This follows from [14], proof of Proposition 1-1-1, since VD( Γ\Sw)=vΰ is a

finite measure. Π

In Theorem 3.2 <Λm can be replaced by any pairwise smallest sufficient sub-field. This theorem generalizes a result of Torgersen [18], p. 47, for dominatedexperiments and of Mussmann [13], Proposition 2.1 and Proposition 2.5, forweakly dominated experiments. The latter paper also gives a characterizationof the smallest sufficient subfield.

Theorem 3,3. If μ is a pivotal measure for 6, then μ=vG for some maximal

orthogonal system G in Lm(G).

Proof. Let D<Σ.Lm(6) be a maximal orthogonal system. Put Sw ={dw/dμ>ty for all vΰ&D. Sw is an (^-support for w. By Lemma 2.3, dzti/dμis <Jίm -measurable. Hence SwξΞ<Jlm and there is a countable set Kw and a pairwisedisjoint family (Swk, k^Kw) in Jίm with Sw= U Swk and 0<μ(Swk)<oo for all

k^JS^tv

k&Kw. By Theorem 3.2, the measures lSwk w, w^D and k^Kw, also definea maximal orthogonal system in Lm(<S). Therefore we shall assume without lossof generality that Q<μ(Sw)<oo holds for all rt^D. Put vx=μ( Γ}Sw) for all

vw^Lm(<S) because of

and Theorem 3.2. Furthermore, vw and w are equivalent for all w^D. There-fore G=(vw, zo^D) is a maximal orthogonal system in Lm(6). It is easily shownthat μ=vG since μ is semi-finite, that is μ(A) = sup {μ(F) : F(Σ.A, F&A, andμ(F)<oo}. Π

EXAMPLE 3.4. Suppose X is the unit interval, <JL the corresponding Borelsets, and 3?= {λ} where X is the Lebesgue measure on X. Then X is also pivotalfor 6 since {0, X} is a smallest sufficient subfield. We have Lm(6)= {v: v=a\

for some real a}. Because λ= Σ λ( Π A ) for anY countable measurable parti-ίej

tion {Aiy ί^J} of X, we see that the pivotal measure λ can be represented as asum of orthogonal measures which are not from Lm(6).

For each P^S let Sp^Jίm be an ̂ -support for P. A subset 3c.JLm is

802 H. LUSCHGY, D. MUSSMANN AND S. YAMADA

called a maximal 6 \ ̂ ^-decomposition if F1Γ\F2^N(3>) for distinct members

F19 F2<=ζFly for each Fe£F there is a PFe5> with PF(F)>Q and F\SPf&N(&),and each B^Jlm such that B\SQ^N(3?} for some Q <Ξ£P and fin^eiV^) forall Fe£F is in N(&). Such an £F exists by Zorn's lemma (cf. [15], p. 171). In

[15], Proposition 3, it is shown that Σ fV( Γl-F) defines a pivotal measure.F<EΞ%

This will also follow from Theorem 3.1 and our next theorem, where the relation

between maximal <?* | (.^-decompositions and maximal orthogonal systems inLm(6) is exhibited. It easily follows from the definition that each maximal£|o?w-decomρosition is a maximal <?* | ̂ ^-decomposition.

Theorem 3.5. The following assertions hold

a) // £? is a maximal 6* | JLm-deconιposition, then {PF( Γ\F):F^3ί} is amaximal orthogonal system in Lm(β).

b) If D is a maximal orthogonal system in Lm(β) and Sw GΞ <Jlm is an β-support for

each w^D, then {Sw: vΰ^D} is a maximal 6*\<Jlm-decomposition.

Proof.a) Let D be a maximal orthogonal system in Lm(6). By Theorem 3.2, for

each F ££? there is an <_τ?w-meesurable gF such that

We conclude that {PF( Γ\F}): F^3} is an orthogonal system in Lm(6).Suppose v &Lm(β)+ and v/\PF( Π F) = 0 for all F e£F. By Theorem 3.2,v=f vD for some o#w-measurable /. We get

0 = ^({/>0>n{lFfo>0}) = ^({/>0}nF) forall FtΞΪF.

The definition of S£ implies ^z?({/>0})=0. Hence v = Q, and the maxi-mality of {PF( Π F ) : Peff} follows.

b) Suppose B^Jίm, B\S^N(&) for some £>eS>*, and J8Π ^eΛΓ^Jfor allw&D. By Theorem 3.2, g( nB)=(l5/) zΊ) for some <^?w-measurable /.We conclude .Q( Γ(B)<=Lm(β) and Q( Γ}B)/\w=Q for all w^D. Max-imality of D implies Q( Γ\B)=Q. Hence B^N(ίP). Now it is easily seen

that {Sw: w^D} is a maximal ίP* | ̂ ^-decomposition. Π

Acknowledgement. We are grateful to Professor D. Plachky makingthis collaboration possible. Π

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MINIMAL L-SPACE AND HALMOS-SAVAGE CRITERION FOR MAJORIZED EXPERIMENTS 803

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H. Luschgy and D. Mussmann

Institute of Mathematical Statistics

University of MϋnsterEinsteinstr. 62

D-4400 Mϋnster

West Germany

S. Yamada

Tokyo University of Fisheries

5-7 Konan 4, Minato-ku

Tokyo 108, Japan